Group #23764667 — subgroups

Subgroups — #23764667

Order 10112.

256 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 69 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#11929614	C₂	1	1	yes	maximal, characteristic
2	#19098590	C₂	1	1	yes	maximal, characteristic
3	#19098571	C₂	1	1	yes	maximal, characteristic
4	C₂×#14579080	C₂	1	1	yes	maximal, characteristic
5	C₂×#14579079	C₂	1	1	yes	maximal, characteristic
6	#19098327	C₂	1	1	yes	maximal, characteristic
7	#419×C₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₅₈	C₂²	1	1	yes	characteristic
9	C₂×#11929614	—	2	2	no	—
10	C₂×#11929614	—	2	2	no	—
11	C₂×#11929614	—	2	2	no	—
12	C₂×#11929608	C₂²	1	1	yes	characteristic
13	C₂×#11929608	C₂²	1	1	yes	characteristic
14	C₂²×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
15	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
16	#14579084	—	2	2	no	—
17	#14579080	—	2	2	no	—
18	#14579079	—	2	2	no	—
19	C₂²×Q₈×C₇₉	C₂²	1	1	yes	characteristic
20	C₂×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
21	#117×C₇₉	C₂²	1	1	yes	characteristic
22	#103×C₇₉	—	2	2	no	—
23	C₂×Dic₁₅₈	—	2	1	no	—
24	C₂×Dic₁₅₈	—	2	1	no	—
25	C₂×Dic₁₅₈	—	2	1	no	—
26	C₂×Dic₁₅₈	—	2	1	no	—
27	C₂×Dic₁₅₈	—	2	1	no	—
28	C₂×Dic₁₅₈	D₄	1	2	yes	—
29	C₂²×Dic₇₉	—	2	1	no	—
30	#11929614	—	2	8	no	—
31	#11929608	—	2	1	no	—
32	#11929608	—	2	1	no	—
33	#11929608	—	2	1	no	—
34	#11929608	—	2	1	no	—
35	C₂×C₇₉⋊C₈	—	2	1	no	—
36	C₂×C₇₉⋊C₈	—	2	1	no	—
37	C₂×C₇₉⋊C₈	—	2	1	no	—
38	C₂×C₇₉⋊C₈	—	2	1	no	—
39	C₂×C₇₉⋊C₈	D₄	1	2	yes	—
40	C₂×Q₈×C₇₉	—	2	1	no	—
41	C₂×Q₈×C₇₉	—	2	1	no	—
42	C₂×Q₈×C₇₉	—	2	1	no	—
43	C₂×Q₈×C₇₉	—	2	1	no	—
44	C₂×Q₈×C₇₉	—	2	1	no	—
45	C₂×Q₈×C₇₉	D₄	1	2	yes	—
46	C₂²×C₄×C₇₉	—	2	1	no	—
47	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
48	M₁₆×C₇₉	—	2	1	no	—
49	M₁₆×C₇₉	—	2	1	no	—
50	C₂×C₈×C₇₉	—	2	1	no	—
51	C₄⋊C₄×C₇₉	D₄	1	2	yes	—
52	C₂²⋊C₄×C₇₉	—	2	1	no	—
53	C₄²×C₇₉	—	2	1	no	—
54	C₂³×C₇₉	#41	1	1	yes	characteristic
55	Q₈×C₇₉	—	2	2	no	—
56	Q₈×C₇₉	—	2	2	no	—
57	Q₈×C₇₉	—	4	1	no	—
58	Q₈×C₇₉	—	4	1	no	—
59	C₂×C₄×C₇₉	—	2	2	no	—
60	C₂×C₄×C₇₉	—	2	2	no	—
61	C₂×C₄×C₇₉	—	4	1	no	—
62	C₂×C₄×C₇₉	—	4	1	no	—
63	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
68	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
69	C₂×Dic₇₉	—	2	2	no	—
70	C₂×Dic₇₉	—	4	1	no	—
71	C₂×Dic₇₉	—	4	1	no	—
72	Dic₁₅₈	—	2	2	no	—
73	Dic₁₅₈	—	2	2	no	—
74	Dic₁₅₈	—	4	1	no	—
75	Dic₁₅₈	—	4	1	no	—
76	C₈×C₇₉	—	4	1	no	—
77	C₇₉⋊C₈	—	2	2	no	—
78	C₇₉⋊C₈	—	4	1	no	—
79	C₇₉⋊C₈	—	4	1	no	—
80	C₂²×C₇₉	—	2	2	no	—
81	C₂²×C₇₉	#121	1	1	yes	characteristic
82	C₂²×C₇₉	#123	1	1	yes	characteristic
83	C₂²×C₇₉	#123	1	1	yes	characteristic
84	C₄×C₇₉	—	4	2	no	—
85	C₄×C₇₉	—	4	2	no	—
86	C₄×C₇₉	#120	1	1	yes	characteristic
87	C₄×C₇₉	#121	1	1	yes	characteristic
88	C₄×C₇₉	#121	1	1	yes	characteristic
89	C₄×C₇₉	#124	1	1	yes	characteristic
90	Dic₇₉	—	4	2	no	—
91	C₂×C₇₉	—	2	1	no	—
92	C₂×C₇₉	—	2	1	no	—
93	C₂×C₇₉	#394	1	1	yes	characteristic
94	C₂×C₇₉	#473	1	2	yes	—
95	#2047	—	79	1	no	maximal, Sylow
96	C₇₉	#2047	1	1	yes	characteristic, complemented, Sylow
97	C₂²×Q₁₆	—	79	1	no	—
98	#424	—	79	1	no	—
99	#419	—	79	1	no	—
100	#419	D₇₉	1	1	yes	characteristic
101	C₂×#101	—	79	1	no	—
102	C₂×#101	—	79	1	no	—
103	#341	—	79	1	no	—
104	C₂²×Q₈	—	79	1	no	—
105	C₂²×Q₈	C₂×D₇₉	1	1	yes	characteristic
106	C₂×Q₁₆	—	158	2	no	—
107	C₂×Q₁₆	—	158	2	no	—
108	C₂×Q₁₆	—	158	2	no	—
109	C₂×M₁₆	—	79	1	no	—
110	C₂×M₁₆	—	79	1	no	—
111	C₂×M₁₆	C₂×D₇₉	1	1	yes	characteristic
112	C₂²×C₈	—	79	1	no	—
113	#117	C₂×D₇₉	1	1	yes	characteristic
114	C₄⋊C₈	—	158	2	no	—
115	#103	—	2	2	no	—
116	#103	—	158	2	no	—
117	#101	—	158	2	no	—
118	#101	—	158	2	no	—
119	C₂×Q₈	—	2	1	no	—
120	C₂×Q₈	—	2	1	no	—
121	C₂×Q₈	—	2	1	no	—
122	C₂×Q₈	—	2	1	no	—
123	C₂×Q₈	—	2	1	no	—
124	C₂×Q₈	—	79	2	no	—
125	C₂×Q₈	—	158	1	no	—
126	C₂×Q₈	—	158	1	no	—
127	C₂×Q₈	—	158	1	no	—
128	C₂×Q₈	—	158	1	no	—
129	C₂×Q₈	—	158	1	no	—
130	C₂×Q₈	D₃₁₆	1	2	yes	—
131	C₂²×C₄	—	2	1	no	—
132	C₂²×C₄	—	158	1	no	—
133	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
134	Q₁₆	—	158	8	no	—
135	M₁₆	—	2	1	no	—
136	M₁₆	—	2	1	no	—
137	M₁₆	—	158	1	no	—
138	M₁₆	—	158	1	no	—
139	M₁₆	—	158	1	no	—
140	M₁₆	—	158	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	79	2	no	—
143	C₂×C₈	—	158	1	no	—
144	C₂×C₈	—	158	1	no	—
145	C₂×C₈	—	158	1	no	—
146	C₂×C₈	—	158	1	no	—
147	C₄⋊C₄	C₂²⋊D₇₉	1	2	yes	—
148	C₂²⋊C₄	—	2	1	no	—
149	C₄²	—	2	1	no	—
150	C₂³	#11929623	1	1	yes	characteristic
151	Q₈	—	2	2	no	—
152	Q₈	—	2	2	no	—
153	Q₈	—	4	1	no	—
154	Q₈	—	4	1	no	—
155	Q₈	—	316	1	no	—
156	Q₈	—	316	1	no	—
157	Q₈	—	158	2	no	—
158	Q₈	—	158	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	1	no	—
162	C₂×C₄	—	4	1	no	—
163	C₂×C₄	—	316	1	no	—
164	C₂×C₄	—	316	1	no	—
165	C₂×C₄	—	158	2	no	—
166	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
167	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
168	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
169	C₂×C₄	C₂×D₃₁₆	1	1	yes	characteristic
170	C₂×C₄	#11929623	1	1	yes	characteristic
171	C₂×C₄	#11929624	1	1	yes	characteristic
172	C₈	—	4	1	no	—
173	C₈	—	316	1	no	—
174	C₈	—	316	1	no	—
175	C₈	—	158	2	no	—
176	C₂²	—	2	2	no	—
177	C₂²	#14579133	1	1	yes	characteristic
178	C₂²	#14579143	1	1	yes	characteristic
179	C₂²	#14579145	1	1	yes	characteristic
180	C₄	—	4	2	no	—
181	C₄	—	4	2	no	—
182	C₄	—	316	2	no	—
183	C₄	#14579131	1	1	yes	characteristic
184	C₄	#14579132	1	1	yes	characteristic
185	C₄	#14579135	1	1	yes	characteristic
186	C₄	#14579147	1	1	yes	characteristic
187	C₂	—	2	1	no	—
188	C₂	—	2	1	no	—
189	C₂	#19098479	1	1	yes	characteristic
190	C₂	#19098836	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11929614

C₂

yes

maximal, characteristic

#19098590

C₂

yes

maximal, characteristic

#19098571

C₂

yes

maximal, characteristic

C₂×#14579080

C₂

yes

maximal, characteristic

C₂×#14579079

C₂

yes

maximal, characteristic

#19098327

C₂

yes

maximal, characteristic

#419×C₇₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₅₈

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₂×#11929608

C₂²

yes

characteristic

C₂²×C₇₉⋊C₈

C₂²

yes

characteristic

C₄⋊(C₇₉⋊C₈)

—

#14579084

—

#14579080

—

#14579079

—

C₂²×Q₈×C₇₉

C₂²

yes

characteristic

C₂×M₁₆×C₇₉

C₂²

yes

characteristic

#117×C₇₉

C₂²

yes

characteristic

#103×C₇₉

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

D₄

yes

—

C₂²×Dic₇₉

—

—

—

—

—

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

D₄

yes

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

D₄

yes

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

M₁₆×C₇₉

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₄⋊C₄×C₇₉

D₄

yes

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₂³×C₇₉

#41

yes

characteristic

Q₈×C₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#394

yes

characteristic

C₂×C₇₉

#473

yes

—

#2047

—

maximal, Sylow

C₇₉

#2047

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

#424

—

#419

—

100

#419

D₇₉

yes

characteristic

101

C₂×#101

—

102

C₂×#101

—

103

#341

—

104

C₂²×Q₈

—

105

C₂²×Q₈

C₂×D₇₉

yes

characteristic

106

C₂×Q₁₆

—

158

—

107

C₂×Q₁₆

—

158

—

108

C₂×Q₁₆

—

158

—

109

C₂×M₁₆

—

110

C₂×M₁₆

—

111

C₂×M₁₆

C₂×D₇₉

yes

characteristic

112

C₂²×C₈

—

113

#117

C₂×D₇₉

yes

characteristic

114

C₄⋊C₈

—

158

—

115

#103

—

116

#103

—

158

—

117

#101

—

158

—

118

#101

—

158

—

119

C₂×Q₈

—

120

C₂×Q₈

—

121

C₂×Q₈

—

122

C₂×Q₈

—

123

C₂×Q₈

—

124

C₂×Q₈

—

125

C₂×Q₈

—

158

—

126

C₂×Q₈

—

158

—

127

C₂×Q₈

—

158

—

128

C₂×Q₈

—

158

—

129

C₂×Q₈

—

158

—

130

C₂×Q₈

D₃₁₆

yes

—

131

C₂²×C₄

—

132

C₂²×C₄

—

158

—

133

C₂²×C₄

C₂²×D₇₉

yes

characteristic

134

Q₁₆

—

158

—

135

M₁₆

—

136

M₁₆

—

137

M₁₆

—

158

—

138

M₁₆

—

158

—

139

M₁₆

—

158

—

140

M₁₆

—

158

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

158

—

144

C₂×C₈

—

158

—

145

C₂×C₈

—

158

—

146

C₂×C₈

—

158

—

147

C₄⋊C₄

C₂²⋊D₇₉

yes

—

148

C₂²⋊C₄

—

149

C₄²

—

150

C₂³

#11929623

yes

characteristic

151

Q₈

—

152

Q₈

—

153

Q₈

—

154

Q₈

—

155

Q₈

—

316

—

156

Q₈

—

316

—

157

Q₈

—

158

—

158

Q₈

—

158

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

316

—

164

C₂×C₄

—

316

—

165

C₂×C₄

—

158

—

166

C₂×C₄

D₄×D₇₉

yes

characteristic

167

C₂×C₄

D₄×D₇₉

yes

characteristic

168

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

169

C₂×C₄

C₂×D₃₁₆

yes

characteristic

170

C₂×C₄

#11929623

yes

characteristic

171

C₂×C₄

#11929624

yes

characteristic

172

C₈

—

173

C₈

—

316

—

174

C₈

—

316

—

175

C₈

—

158

—

176

C₂²

—

177

C₂²

#14579133

yes

characteristic

178

C₂²

#14579143

yes

characteristic

179

C₂²

#14579145

yes

characteristic

180

C₄

—

181

C₄

—

182

C₄

—

316

—

183

C₄

#14579131

yes

characteristic

184

C₄

#14579132

yes

characteristic

185

C₄

#14579135

yes

characteristic

186

C₄

#14579147

yes

characteristic

187

C₂

—

188

C₂

—

189

C₂

#19098479

yes

characteristic

190

C₂

#19098836

yes

—

	subgroup	G/H
center	C₂²	#14579133
commutator, cosocle	C₂²×C₄×C₇₉	C₂³
Frattini	C₂²×C₄	C₂²×D₇₉
Fitting	#419×C₇₉	C₂
socle	C₂²×C₇₉	#121
prime radical	C₂³×C₇₉	#41
hypercenter	#419	D₇₉
hypocenter	C₇₉	#2047
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #419

subgroup

G/H

center

C₂²

#14579133

commutator, cosocle

C₂²×C₄×C₇₉

C₂³

Frattini

C₂²×C₄

C₂²×D₇₉

Fitting

#419×C₇₉

C₂

C₂²×C₇₉

C₂³×C₇₉

D₇₉

C₇₉

G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #419

N	H ≅ G/N	type
96. C₇₉	#2047	semidirect

H ≅ G/N

type

96. C₇₉

#2047

semidirect

Subgroups — #23764667

Conjugacy classes of subgroups

Complements